| L(s) = 1 | − 7.56·2-s + 25.2·4-s − 40.1·5-s + 194.·7-s + 51.1·8-s + 303.·10-s + 39.1·11-s + 705.·13-s − 1.47e3·14-s − 1.19e3·16-s − 1.22e3·17-s + 1.89e3·19-s − 1.01e3·20-s − 296.·22-s − 529·23-s − 1.51e3·25-s − 5.33e3·26-s + 4.91e3·28-s + 8.57e3·29-s − 9.58e3·31-s + 7.40e3·32-s + 9.24e3·34-s − 7.82e3·35-s − 8.14e3·37-s − 1.43e4·38-s − 2.05e3·40-s − 4.83e3·41-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 0.788·4-s − 0.718·5-s + 1.50·7-s + 0.282·8-s + 0.960·10-s + 0.0976·11-s + 1.15·13-s − 2.01·14-s − 1.16·16-s − 1.02·17-s + 1.20·19-s − 0.566·20-s − 0.130·22-s − 0.208·23-s − 0.484·25-s − 1.54·26-s + 1.18·28-s + 1.89·29-s − 1.79·31-s + 1.27·32-s + 1.37·34-s − 1.07·35-s − 0.978·37-s − 1.60·38-s − 0.203·40-s − 0.449·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.001769346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.001769346\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 + 7.56T + 32T^{2} \) |
| 5 | \( 1 + 40.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 194.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 39.1T + 1.61e5T^{2} \) |
| 13 | \( 1 - 705.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.22e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.89e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 8.57e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.58e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.14e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.83e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.06e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 222.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.57e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.07e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 8.69e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.83e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.02e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.06e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.56e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.76e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.30e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.18668595484146847904399029080, −10.68248205882521844774470719248, −9.312412793407084876323455826022, −8.411825045445811253764276753905, −7.911152367323532309887212830921, −6.85565860905127715612747454050, −5.11717980872924183057368713393, −3.92463888687786528694823653740, −1.88049032196043731928189818182, −0.78800811617662791888698363201,
0.78800811617662791888698363201, 1.88049032196043731928189818182, 3.92463888687786528694823653740, 5.11717980872924183057368713393, 6.85565860905127715612747454050, 7.911152367323532309887212830921, 8.411825045445811253764276753905, 9.312412793407084876323455826022, 10.68248205882521844774470719248, 11.18668595484146847904399029080
